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Definition

If CC and DD are monoidal categories, an oplax monoidal functorF:C→DF : C \to D is defined to be a lax monoidal functorF:Cop→DopF: C^{op} \to D^{op}. So, among other things, tensor products are preserved up to morphisms of the following sort in DD:

Properties

An oplax monoidal functor sends comonoids in CC to comonoids in DD, just as a lax monoidal functor sends monoids in CC to monoids in DD. For this reason an oplax monoidal functor is sometimes called a lax comonoidal functor. The other obvious terms, colax monoidal and lax opmonoidal, also exist (or at least are attested on Wikipedia).